Then Γ0(N) is a discrete subgroup of G. Note that Γ0(1) = SL2(Z). Now we make three choices that determine our space of modular forms. We pick an integer k, called the weight, a positive integer N , called the level, and a Dirichlet character χ : Z/NZ → C, called the nebentypus. If you’ve never seen modular forms before, you should immediately specialize to the easiest case N = 1, in which case...

For this paper we assume familiarity with the basics of the theory of modular forms as may be found, for instance, in Serre’s classic introduction [12]. A weakly holomorphic modular form of weight k ∈ 2Z for Γ = PSL2(Z) is a holomorphic function f on the upper half-plane that satisfies f( cτ+d ) = (cτ + d)f(τ) for all ( a b c d ) ∈ Γ and that has a q-expansion of the form f(τ) = ∑ n≥n0 a(n)q , ...

background with the advent of hystroscopic surgery, abnormalities confined to the uterine cavity such as endometrial polyps, submucous myomas, uterine septae and synechia , were supposed to be treated effectively. diagnostic hysteroscopy is the gold standard for investigating the intrauterine diseases as it allows for biopsy and removal of lesions. objectives forty premenopausal women without t...

A dilatation structure on a metric space, is a notion in between a group and a differential structure. The basic objects of a dilatation structure are dilatations (or contractions). The axioms of a dilatation structure set the rules of interaction between different dilatations. There are two notions of linearity associated to dilatation structures: the linearity of a function between two dilata...